reserve E, x, y, X for set;
reserve A, B, C, D for Subset of E^omega;
reserve a, a1, a2, b, c, c1, c2, d, ab, bc for Element of E^omega;
reserve e for Element of E;
reserve i, j, k, l, n, n1, n2, m for Nat;

theorem Th57:
  A ^^ (A*) = (A*) ^^ A
proof
A1: A* = {<%>E} \/ A ^^ (A*) & A* = {<%>E} \/ (A*) ^^ A by Th56;
  now
    per cases;
    suppose
A2:   <%>E in A;
      then A* = A ^^ (A*) by Th54;
      hence thesis by A2,Th54;
    end;
    suppose
A3:   not <%>E in A;
      then not <%>E in (A*) ^^ A by Th15;
      then
A4:   {<%>E} misses (A*) ^^ A by ZFMISC_1:50;
      not <%>E in A ^^ (A*) by A3,Th15;
      then {<%>E} misses A ^^ (A*) by ZFMISC_1:50;
      hence thesis by A1,A4,XBOOLE_1:71;
    end;
  end;
  hence thesis;
end;
